Q: A plumber stands on the end of a wrench handle, applying a force of $$900 N at a $$distance of $$0.80 m $$from the axis, at an angle of $$19^ ext{o}$$ with the horizontal. What is the magnitude of the torque produced?

$$\tau = rF\sin\phi = 0.80 \times 900 \times \sin(19^ ext{o})$$

Q: What is the vector formula for torque $$\vec{\tau}$$ produced by a force $$\vec{F}$$ applied at position $$\vec{r}$$ relative to the axis?

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Question 1

A rigid body rotates about a fixed axis with angular speed $$\omega. $$Its moment of inertia about this axis is $$I. $$What is the formula for its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2}I\$$omega^2$$

Question 2

Why is it easier to start rotating an apparatus when its mass is close to the axis of rotation?

Accepted Answer

Because the moment of inertia is smaller, requiring less torque to achieve the same angular acceleration.

Question 3

A hummingbird can flap its wings up to 70 beats per second, while an Andean condor flaps at about one beat per second. What physical property explains this difference?

Accepted Answer

The hummingbird has a smaller moment of inertia, allowing faster wing movement.

Question 4

Which formula gives the moment of inertia for a solid cylinder of mass $$M$$ and radius $$R$$ rotating about its central axis?

Accepted Answer

$$I = \frac{1}{2}MR^2$

Question 5

A machine part consists of three disks linked by lightweight struts. Disk A ($$m_A = 0.30 kg$$), Disk B ($$m_B = 0.10 kg$$), and Disk C ($$m_C = 0.20 kg) $$are positioned as shown. What is the moment of inertia about the axis through disk A, perpendicular to the plane?

Accepted Answer

$$I = I_A + m_B d_{AB}^2 + m_C d_{AC}^2$$

Question 6

In the scenario where a cable unwinds from a solid cylinder of mass $$M$$ and radius $$R$$, and a block of mass $$m$$ falls from height $$h$$, what is the expression for the speed $$v of $$the block as it strikes the floor?

Accepted Answer

$$v = \sqrt{\frac{2mgh}{m + \frac{1}{2}M}}$$

Question 7

A plumber stands on the end of a wrench handle, applying a force of $$900 N at a $$distance of $$0.80 m $$from the axis, at an angle of $$19^	ext{o}$$ with the horizontal. What is the magnitude of the torque produced?

Accepted Answer

$$	au = rF\sin\phi = 0.80 	imes 900 	imes \sin(19^	ext{o})$$

Question 8

Which force applied to a wrench is most effective for loosening a bolt?

Accepted Answer

A force applied perpendicular to the wrench and farthest from the axis of rotation.

Question 9

What is the vector formula for torque $$\vec{	au}$$ produced by a force $$\vec{F}$$ applied at position $$\vec{r}$$ relative to the axis?

Accepted Answer

$$\vec{	au} = \vec{r} 	imes \vec{F}$$

Question 10

A child pushes tangentially on a merry-go-round. Which component of force contributes to the torque?

Accepted Answer

The tangential component of the force.

Question 11

What is the rotational analog of Newton’s second law for a rigid body?

Accepted Answer

$$	au = I\alpha$$

Question 12

A helicopter’s main rotor spins at a constant rate. Positive work is done by the engine, negative work by air resistance. What is the net work done on the rotor?

Accepted Answer

Zero, because the kinetic energy remains constant.

Question 13

What is the formula for the angular momentum $$\vec{L} of a $$rigid body rotating about a symmetry axis?

Accepted Answer

$$\vec{L} = I\vec{\omega}$$

Question 14

If the net external torque acting on a system is zero, what happens to the total angular momentum?

Accepted Answer

It remains constant (is conserved).

Question 15

A professor spins on a turntable holding dumbbells. When he pulls the dumbbells inward, what happens to his angular velocity?

Accepted Answer

It increases, because his moment of inertia decreases and angular momentum is conserved.

Question 16

For an extended body to be in static equilibrium, which two conditions must be satisfied?

Accepted Answer

The sum of all external forces and the sum of all external torques must be zero.

Question 17

In high jump, why does arching the body so that the center of mass passes under the bar require less energy than straddling the bar?

Accepted Answer

Because the increase in gravitational potential energy is smaller when the center of mass passes under the bar.

Question 18

What is the formula for the gravitational potential energy of an extended body?

Accepted Answer

$$U_{grav} = Mg y_{cm}$$

Question 19

A block of mass $$m$$ falls while unwinding a cable from a cylinder of mass $$M$$ and radius $$R. $$What is the angular speed $$\omega of $$the cylinder as the block strikes the floor?

Accepted Answer

$$\omega = \frac{v}{R}$$, where $$v is $$the speed of the block.

Rotational Motion and Dynamics: Study Notes

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